It is given that the straight line L1: (k+1) x-y-8=0 is parallel to the straight line L2: (2k-1) x-y-3k=0.

1) find k

2) if the straight line L3 is perpendicular to L2 and has the same x-intercept as L2, find the equation of L3

Part 1: k = 2

Part 2: y = (-1/3) x - 6

Step-by-step explanation:

Hello!

Recall that parallel lines have the same slope. Given (k+1) x-y-8=0 and (2k-1) x-y-3k=0, we'll solve each equation for its slope in terms of k and then equate the two slopes to determine the value of k.

We can immediately read off the slope of (2k-1) x-y-3k=0: It's the coefficient, 2k-1, of x. Similarly, the slope of the other line is k+1. Equating these quantities, we get:

2k - 1 = k + 1. Then k = 2, and the slope of the parallel lines is 2 (2) - 1, or 3.

Part 2: A line perpendicular to these parallel lines has a slope of - 1/3, which is the negative reciprocal of 3.

Let's look at L2, substitute 2 for k and find the equation of the line:

(2k-1) x-y-3k=0 = > (2[2]-1) x-y-3[2]=0, or 3x - y - 6 = 0, or y = 3x - 6. Here, the y-intercept is - 6.

Finally, let's write the equation of L3: It has the slope - 1/3 and the y-intercept of - 6:

y = (-1/3) x - 6